1.Download free ebooks at bookboon.com 2 Chris Long & Naser Sayma Heat Transfer

2. Download free ebooks at bookboon.com 3 Heat Transfer 2009 Chris Long, Naser Sayma & Ventus Publishing ApS ISBN 978-87-7681-432-8 3. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 4 Contents 1. Introduction 1.1 Heat Transfer Modes 1.2 System of Units 1.3 Conduction 1.4 Convection 1.5 Radiation 1.6 Summary 1.7 Multiple Choice assessment 2. Conduction 2.1 The General Conduction Equation 2.2 One-Dimensional Steady-State Conduction in Radial Geometries: 2.3 Fins and Extended Surfaces 2.4 Summary 2.5 Multiple Choice Assessment 3. Convection 3.1 The convection equation 3.2 Flow equations and boundary layer 3.3 Dimensional analysis 6 6 7 7 12 16 20 21 25 25 34 38 48 48 58 58 60 70 Contents 4. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 5 3.4 Forced Convection relations 3.5 Natural convection 3.6 Summary 3.7 Multiple Choice Assessment 4. Radiation 4.1 Introduction 4.2 Radiative Properties 4.3 Kirchhoffs law of radiation 4.4 View factors and view factor algebra 4.5 Radiative Exchange Between a Number of Grey Surfaces 4.6 Radiation Exchange Between Two Grey Bodies 4.7 Summary 4.8 Multiple Choice Assessment 5. Heat Exchangers 5.1 Introduction 5.2 Classication of Heat Exchangers 5.4 Analysis of Heat Exchangers 5.5 Summary 5.6 Multiple Choice Assessment References 76 90 100 101 107 107 109 111 111 115 121 122 123 127 127 128 139 152 153 156 Contents Bernard chose a career at KPMG. He nds himself invol- ved in a diverse range of pro- jects which enable him to see behind the scenes at various listed companies. This dynamic working environment inspires him both on a professional level and through the people he works with. Inspiring careers for inspiring people. 5. Download free ebooks at bookboon.com Heat Transfer 6 Introduction 1. Introduction Energy is defined as the capacity of a substance to do work. It is a property of the substance and it can be transferred by interaction of a system and its surroundings. The student would have encountered these interactions during the study of Thermodynamics. However, Thermodynamics deals with the end states of the processes and provides no information on the physical mechanisms that caused the process to take place. Heat Transfer is an example of such a process. A convenient definition of heat transfer is energy in transition due to temperature differences. Heat transfer extends the Thermodynamic analysis by studying the fundamental processes and modes of heat transfer through the development of relations used to calculate its rate. The aim of this chapter is to console existing understanding and to familiarise the student with the standard of notation and terminology used in this book. It will also introduce the necessary units. 1.1 Heat Transfer Modes The different types of heat transfer are usually referred to as modes of heat transfer. There are three of these: conduction, convection and radiation. Conduction: This occurs at molecular level when a temperature gradient exists in a medium, which can be solid or fluid. Heat is transferred along that temperature gradient by conduction. Convection: Happens in fluids in one of two mechanisms: random molecular motion which is termed diffusion or the bulk motion of a fluid carries energy from place to place. Convection can be either forced through for example pushing the flow along the surface or natural as that which happens due to buoyancy forces. Radiation: Occurs where heat energy is transferred by electromagnetic phenomenon, of which the sun is a particularly important source. It happens between surfaces at different temperatures even if there is no medium between them as long as they face each other. In many practical problems, these three mechanisms combine to generate the total energy flow, but it is convenient to consider them separately at this introductory stage. We need to describe each process symbolically in an equation of reasonably simple form, which will provide the basis for subsequent calculations. We must also identify the properties of materials, and other system characteristics, that influence the transfer of heat. 6. Download free ebooks at bookboon.com Heat Transfer 7 Introduction 1.2 System of Units Before looking at the three distinct modes of transfer, it is appropriate to introduce some terms and units that apply to all of them. It is worth mentioning that we will be using the SI units throughout this book: The rate of heat flow will be denoted by the symbolQ . It is measured in Watts (W) and multiples such as (kW) and (MW). It is often convenient to specify the flow of energy as the heat flow per unit area which is also known as heat flux. This is denoted by q . Note that, AQq / where A is the area through which the heat flows, and that the units of heat flux are (W/m2). Naturally, temperatures play a major part in the study of heat transfer. The symbol T will be used for temperature. In SI units, temperature is measured in Kelvin or Celsius: (K) and ( C). Sometimes the symbol t is used for temperature, but this is not appropriate in the context of transient heat transfer, where it is convenient to use that symbol for time. Temperature difference is denoted in Kelvin (K). The following three subsections describe the above mentioned three modes of heat flow in more detail. Further details of conduction, convection and radiation will be presented in Chapters 2, 3 and 4 respectively. Chapter 5 gives a brief overview of Heat Exchangers theory and application which draws on the work from the previous Chapters. 1.3 Conduction The conductive transfer is of immediate interest through solid materials. However, conduction within fluids is also important as it is one of the mechanisms by which heat reaches and leaves the surface of a solid. Moreover, the tiny voids within some solid materials contain gases that conduct heat, albeit not very effectively unless they are replaced by liquids, an event which is not uncommon. Provided that a fluid is still or very slowly moving, the following analysis for solids is also applicable to conductive heat flow through a fluid. 7. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 8 Introduction Figure 1.1 shows, in schematic form, a process of conductive heat transfer and identifies the key quantities to be considered: Q : the heat flow by conduction in the x- direction (W) A : the area through which the heat flows, normal to the x-direction (m2) Figure 1-1: One dimensional conduction Bereit fr neue Wege? Nordafrika Bestimmung einer neuen Landwirtschaftsstrategie fr ein Entwicklungsland. Herausforderungen gab es viele, sowohl in wirtschaftlicher, sozialer, ko- logischer als auch in politischer Hinsicht. McKin- sey hat Lsungen erarbeitet, die das Wachstum der Landwirtschaft dank Produkten sichern, fr die eine grosse Nachfrage besteht und die hohe Gewinne einbringen. Ausserdem wurden lokale Initiativen fr den Kampf gegen die Armut in lndlichen Gegenden ins Leben gerufen. Schliessen Sie sich uns an. www.mckinsey.ch creatives.com 8. Download free ebooks at bookboon.com Heat Transfer 9 Introduction dx dT : the temperature gradient in the x-direction (K/m) These quantities are related by Fourier's Law, a model proposed as early as 1822: dx dT kq dx dT Ak-=Q -=or 1 (1.1) A significant feature of this equation is the negative sign. This recognises that the natural direction for the flow of heat is from high temperature to low temperature, and hence down the temperature gradient. The additional quantity that appears in this relationship is k , the thermal conductivity (W/m K) of the material through which the heat flows. This is a property of the particular heat-conducting substance and, like other properties, depends on the state of the material, which is usually specified by its temperature and pressure. The dependence on temperature is of particular importance. Moreover, some materials such as those used in building construction are capable of absorbing water, either in finite pores or at the molecular level, and the moisture content also influences the thermal conductivity. The units of thermal conductivity have been determined from the requirement that Fourier's law must be dimensionally consistent. Considering the finite slab of material shown in Figure 1.1, we see that for one-dimensional conduction the temperature gradient is: L - TT = d x d T 12 2 Hence for this situation the transfer law can also be written L TT kq L T-TAk=Q 21 21 - =or 3 (1.2) = C k (1.3)4 Table 1.1 gives the values of thermal conductivity of some representative solid materials, for conditions of normal temperature and pressure. Also shown are values of another property characterising the flow of heat through materials, thermal diffusivity, which is related to the conductivity by: Where is the density in 3 / mkg of the material and C its specific heat capacity in KkgJ / . 9. Download free ebooks at bookboon.com Heat Transfer 10 Introduction The thermal diffusivity indicates the ability of a material to transfer thermal energy relative to its ability to store it. The diffusivity plays an important role in unsteady conduction, which will be considered in Chapter 2. As was noted above, the value of thermal conductivity varies significantly with temperature, even over the range of climatic conditions found around the world, let alone in the more extreme conditions of cold-storage plants, space flight and combustion. For solids, this is illustrated by the case of mineral wool, for which the thermal conductivity might change from 0.04 to 0.28 W/m K across the range 35 to - 35 C. Table 1-1 Thermal conductivity and diffusivity for typical solid materials at room temperature Material k W/m K mm2/s Material k W/m K mm2/s Copper Aluminium Mild steel Polyethylene Face Brick 350 236 50 0.5 1.0 115 85 13 0.15 0.75 Medium concrete block Dense plaster Stainless steel Nylon, Rubber Aerated concrete 0.5 0.5 14 0.25 0.15 0.35 0.40 4 0.10 0.40 Glass Fireclay brick Dense concrete Common brick 0.9 1.7 1.4 0.6 0.60 0.7 0.8 0.45 Wood, Plywood Wood-wool slab Mineral wool expanded Expanded polystyrene 0.15 0.10 0.04 0.035 0.2 0.2 1.2 1.0 For gases the thermal conductivities can vary significantly with both pressure and temperature. For liquids, the conductivity is more or less insensitive to pressure. Table 1.2 shows the thermal conductivities for typical gases and liquids at some given conditions. Table 1-2 Thermal conductivity for typical gases and liquids Material k [W/m K] Gases Argon (at 300 K and 1 bar) Air (at 300 K and 1 bar) Air (at 400 K and 1 bar) Hydrogen (at 300 K and 1 bar) Freon 12 (at 300 K 1 bar) 0.018 0.026 0.034 0.180 0.070 Liquids Engine oil (at 20oC) Engine oil (at 80oC) Water (at 20oC) Water (at 80oC) 0.145 0.138 0.603 0.670 10. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 11 Introduction Mercury(at 27oC) 8.540 Note the very wide range of conductivities encountered in the materials listed in Tables 1.1 and 1.2. Some part of the variability can be ascribed to the density of the materials, but this is not the whole story (Steel is more dense than aluminium, brick is more dense than water). Metals are excellent conductors of heat as well as electricity, as a consequence of the free electrons within their atomic lattices. Gases are poor conductors, although their conductivity rises with temperature (the molecules then move about more vigorously) and with pressure (there is then a higher density of energy-carrying molecules). Liquids, and notably water, have conductivities of intermediate magnitude, not very different from those for plastics. The low conductivity of many insulating materials can be attributed to the trapping of small pockets of a gas, often air, within a solid material which is itself a rather poor conductor. Example 1.1 Calculate the heat conducted through a 0.2 m thick industrial furnace wall made of fireclay brick. Measurements made during steady-state operation showed that the wall temperatures inside and outside the furnace are 1500 and 1100 K respectively. The length of the wall is 1.2m and the height is 1m. 11. Download free ebooks at bookboon.com Heat Transfer 12 Introduction Solution We first need to make an assumption that the heat conduction through the wall is one dimensional. Then we can use Equation 1.2: L TT AkQ 12 The thermal conductivity for fireclay brick obtained from Table 1.1 is 1.7 W/m K The area of the wall 2 m2.10.12.1A Thus: W4080 m2.0 K1100K1500 m2.1KW/m7.1 2 Q Comment: Note that the direction of heat flow is from the higher temperature inside to the lower temperature outside. 1.4 Convection Convection heat transfer occurs both due to molecular motion and bulk fluid motion. Convective heat transfer may be categorised into two forms according to the nature of the flow: natural Convection and forced convection. In natural of free convection, the fluid motion is driven by density differences associated with temperature changes generated by heating or cooling. In other words, fluid flow is induced by buoyancy forces. Thus the heat transfer itself generates the flow which conveys energy away from the point at which the transfer occurs. In forced convection, the fluid motion is driven by some external influence. Examples are the flows of air induced by a fan, by the wind, or by the motion of a vehicle, and the flows of water within heating, cooling, supply and drainage systems. In all of these processes the moving fluid conveys energy, whether by design or inadvertently. 12. Download free ebooks at bookboon.com Heat Transfer 13 Introduction Floor Ceiling Radiator Free convection cell Solid surface Natural convection Forced convection Figure 1-2: Illustration of the process of convective heat transfer The left of Figure 1.2 illustrates the process of natural convective heat transfer. Heat flows from the radiator to the adjacent air, which then rises, being lighter than the general body of air in the room. This air is replaced by cooler, somewhat denser air drawn along the floor towards the radiator. The rising air flows along the ceiling, to which it can transfer heat, and then back to the lower part of the room to be recirculated through the buoyancy-driven cell of natural convection. The word radiator has been written above in that way because the heat transfer from such devices is not predominantly through radiation; convection is important as well. In fact, in a typical central heating radiator approximately half the heat transfer is by (free) convection. The right part of Figure 1.2 illustrates a process of forced convection. Air is forced by a fan carrying with it heat from the wall if the wall temperature is lower or giving heat to the wall if the wall temperature is lower than the air temperature. If 1T is the temperature of the surface receiving or giving heat, and T is the average temperature of the stream of fluid adjacent to the surface, then the convective heat transfer Q is governed by Newtons law: )-(=or 21 TThq)T-T(Ah=Q c21c 5 (1.3) Another empirical quantity has been introduced to characterise the convective transfer mechanism. This is hc, the convective heat transfer coefficient, which has units [W/m2 K]. This quantity is also known as the convective conductance and as the film coefficient. The term film coefficient arises from a simple, but not entirely unrealistic, picture of the process of convective heat transfer at a surface. Heat is imagined to be conducted through a thin stagnant film of fluid at the surface, and then to be convected away by the moving fluid beyond. Since the 13. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 14 Introduction fluid right against the wall must actually be at rest, this is a fairly reasonable model, and it explains why convective coefficients often depend quite strongly on the conductivity of the fluid. Table 1-3 Representative range of convective heat transfer coefficient Nature of Flow Fluid hc [W/m2 K] Surfaces in buildings Air 1 - 5 Surfaces outside buildings Air 5-150 Across tubes Gas Liquid 10 - 60 60 - 600 In tubes Gas Organic liquid Water Liquid metal 60 - 600 300 - 3000 600 - 6000 6000 - 30000 Natural convection Gas Liquid 0.6 - 600 60 - 3000 Condensing Liquid film Liquid drops 1000 - 30000 30000 - 300000 Boiling Liquid/vapour 1000 - 10000 Untersttzung fr Ihre Seminararbeit! Die PwC-Online-Bibliothek bietet ein groes Archiv an wichtigen Studien und Fachbeitrgen. Weitere Informationen unter www.pwc-career.de 786 14. Download free ebooks at bookboon.com Heat Transfer 15 Introduction The film coefficient is not a property of the fluid, although it does depend on a number of fluid properties: thermal conductivity, density, specific heat and viscosity. This single quantity subsumes a variety of features of the flow, as well as characteristics of the convecting fluid. Obviously, the velocity of the flow past the wall is significant, as is the fundamental nature of the motion, that is to say, whether it is turbulent or laminar. Generally speaking, the convective coefficient increases as the velocity increases. A great deal of work has been done in measuring and predicting convective heat transfer coefficients. Nevertheless, for all but the simplest situations we must rely upon empirical data, although numerical methods based on computational fluid dynamics (CFD) are becoming increasingly used to compute the heat transfer coefficient for complex situations. Table 1.3 gives some typical values; the cases considered include many of the situations that arise within buildings and in equipment installed in buildings. Example 1.2 A refrigerator stands in a room where the air temperature is 20oC. The surface temperature on the outside of the refrigerator is 16oC. The sides are 30 mm thick and have an equivalent thermal conductivity of 0.1 W/m K. The heat transfer coefficient on the outside 9 is 10 W/m2K. Assuming one dimensional conduction through the sides, calculate the net heat flow and the surface temperature on the inside. Solution Let isT , and osT , be the inside surface and outside surface temperatures, respectively and fT the fluid temperature outside. The rate of heat convection per unit area can be calculated from Equation 1.3: )( , fos TThq 2 W/m40)2016(10q This must equal the heat conducted through the sides. Thus we can use Equation 1.2 to calculate the surface temperature: L TT kq isos ,, 15. Download free ebooks at bookboon.com Heat Transfer 16 Introduction 03.0 16 1.040 ,isT CT is 4, Comment: This example demonstrates the combination of conduction and convection heat transfer relations to establish the desired quantities. 1.5 Radiation While both conductive and convective transfers involve the flow of energy through a solid or fluid substance, no medium is required to achieve radiative heat transfer. Indeed, electromagnetic radiation travels most efficiently through a vacuum, though it is able to pass quite effectively through many gases, liquids and through some solids, in particular, relatively thin layers of glass and transparent plastics. Figure 1.3 indicates the names applied to particular sections of the electromagnetic spectrum where the band of thermal radiation is also shown. This includes: the rather narrow band of visible light; the wider span of thermal radiation, extending well beyond the visible spectrum. Our immediate interest is thermal radiation. It is of the same family as visible light and behaves in the same general fashion, being reflected, refracted and absorbed. These phenomena are of particular importance in the calculation of solar gains, the heat inputs to buildings from the sun and radiative heat transfer within combustion chambers. Figure 1-3: Illustration of electromagnetic spectrum 16. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 17 Introduction It is vital to realise that every body, unless at the absolute zero of temperature, both emits and absorbs energy by radiation. In many circumstances the inwards and outwards transfers nearly cancel out, because the body is at about the same temperature as its surroundings. This is your situation as you sit reading these words, continually exchanging energy with the surfaces surrounding you. In l884 Boltzmann put forward an expression for the net transfer from an idealised body (Black body) with surface area A1 at absolute temperature T1 to surroundings at uniform absolute temperature T2: )-(=or 4 2 4 1 TTq)T-T(A=Q 4 2 4 11 6 (1.4) with the Stefan-Boltzmann constant, which has the value 5.67 x 10-8 W/m2 K4 and T [K] = T [ C] + 273 is the absolute temperature. The bodies considered above are idealised, in that they perfectly absorb and emit radiation of all wave-lengths. The situation is also idealised in that each of the bodies that exchange radiation has a uniform surface temperature. A development of Boltzmann's law which allows for deviations from this pattern is eine Marke von MSW & Partner Haben Sie das Berater-Gen? www.career-venture.de twitter.com/CareerVenture facebook.com/CareerVenture business & consulting 24. September 2012 Frankfurt am Main Bewerbungsschluss: 29.08.2012 Teilnehmende Unternehmen 17. Download free ebooks at bookboon.com Heat Transfer 18 Introduction )T-T(AF=Q 4 2 4 1112 7 (1.5) With the emissivity, or emittance, of the surface A1, a dimensionless factor in the range 0 to 1, F12 is the view factor, or angle factor, giving the fraction of the radiation from A1 that falls on the area A2 at temperature T2, and therefore also in the range 0 to 1. Another property of the surface is implicit in this relationship: its absorbtivity. This has been taken to be equal to the emissivity. This is not always realistic. For example, a surface receiving short-wave-length radiation from the sun may reject some of that energy by re-radiation in a lower band of wave-lengths, for which the emissivity is different from the absorbtivity for the wave-lengths received. The case of solar radiation provides an interesting application of this equation. The view factor for the Sun, as seen from the Earth, is very small; despite this, the very high solar temperature (raised to the power 4) ensures that the radiative transfer is substantial. Of course, if two surfaces do not see one another (as, for instance, when the Sun is on the other side of the Earth), the view factor is zero. Table 1.4 shows values of the emissivity of a variety of materials. Once again we find that a wide range of characteristics are available to the designer who seeks to control heat transfers. The values quoted in the table are averages over a range of radiation wave-lengths. For most materials, considerable variations occur across the spectrum. Indeed, the surfaces used in solar collectors are chosen because they possess this characteristic to a marked degree. The emissivity depends also on temperature, with the consequence that the radiative heat transfer is not exactly proportional to T3. An ideal emitter and absorber is referred to as a black body, while a surface with an emissivity less than unity is referred to as grey. These are somewhat misleading terms, for our interest here is in the infra-red spectrum rather than the visible part. The appearance of a surface to the eye may not tell us much about its heat-absorbing characteristics. Table 1-4 Representative values of emissivity Ideal black body White paint Gloss paint Brick Rusted steel 1.00 0.97 0.9 0.9 0.8 Aluminium paint Galvanised steel Stainless steel Aluminium foil Polished copper Perfect mirror 0.5 0.3 0.15 0.12 0.03 0 18. Download free ebooks at bookboon.com Heat Transfer 19 Introduction Although it depends upon a difference in temperature, Boltzmann's Law (Equations 1.4, 1.5) does not have the precise form of the laws for conductive and convective transfers. Nevertheless, we can make the radiation law look like the others. We introduce a radiative heat transfer coefficient or radiative conductance through Q )T-T(Ah= 211r 8 (1.6) Comparison with the developed form of the Boltzmann Equation (1.5), plus a little algebra, gives )T+T()T+T(F= )T-T(A Q =h 2 2 2 12112 211 r 9 If the temperatures of the energy-exchanging bodies are not too different, this can be approximated by TF4=h 3 av12r 10 (1.7) where Tav is the average of the two temperatures. Obviously, this simplification is not applicable to the case of solar radiation. However, the temperatures of the walls, floor and ceiling of a room generally differ by only a few degrees. Hence the approximation given by Equation (1.7) is adequate when transfers between them are to be calculated. Example 1.3 Surface A in the Figure is coated with white paint and is maintained at temperature of 200oC. It is located directly opposite to surface B which can be considered a black body and is maintained ate temperature of 800oC. Calculate the amount of heat that needs to be removed from surface A per unit area to maintain its constant temperature. 19. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 20 Introduction A 200o C B 800o C Solution The two surfaces are assumed to be infinite and close to each other that they are only exchanging heat with each other. The view factor can then assumed to be 1. The heat gained by surface A by radiation from surface B can be computed from Equation 1.5: )-(= 44 ABAB TTFq The emissivity of white coated paint is 0.97 from Table 1.4 Thus 2448 W/m71469)400-1073(11067.597.0=q This amount of heat needs to be removed from surface A by other means such as conduction, convection or radiation to other surfaces to maintain its constant temperature. 1.6 Summary This chapter introduced some of the basic concepts of heat transfer and indicates their significance in the context of engineering applications. ALEXANDER WOLLTE MEHR VERANTWORTUNG. WIR VERTRAUTEN IHM UNSERE WICHTIGSTEN KUNDEN AN. Als Alexander mit der Betreuung seiner neuen Private Banking Kunden startete, war nicht abzusehen, wohin ihn sein Engagement fhren wrde. Heute bert er 50 unserer grssten Kunden in Nahost. Seine Empfehlungen sind fr die Kunden sehr wertvoll die Erfahrung fr ihn unbezahlbar. Lesen Sie Alexanders Geschichte unter credit-suisse.com/careers LOCATION: ZURICH 20. Download free ebooks at bookboon.com Heat Transfer 21 Introduction We have seen that heat transfer can occur by one of three modes, conduction, convection and radiation. These often act together. We have also described the heat transfer in the three forms using basic laws as follows: Conduction: ]W[ dx dT kA=Q Where thermal conductivity k [W/m K] is a property of the material Convection from a surface: ]W[)( TThA=Q s Where the convective coefficient h [W/m2 K] depends on the fluid properties and motion. Radiation heat exchange between two surfaces of temperatures 1T and 2T : )T-T(AF=Q 4 2 4 1112 Where is the Emissivity of surface 1 and F12 is the view factor. Typical values of the relevant material properties and heat transfer coefficients have been indicated for common materials used in engineering applications. 1.7 Multiple Choice assessment 1. The units of heat flux are: Watts Joules Joules / meters2 Watts / meters2 Joules / Kg K 2. The units of thermal conductivity are: Watts / meters2 K Joules Joules / meters2 Joules / second meter K Joules / Kg K 21. Download free ebooks at bookboon.com Heat Transfer 22 Introduction 3. The heat transfer coefficient is defined by the relationship h = m Cp T h = k / L h = q / T h = Nu k / L h = Q / T 4. Which of these materials has the highest thermal conductivity ? air water mild steel titanium aluminium 5. Which of these materials has the lowest thermal conductivity ? air water mild steel titanium aluminium 6. In which of these is free convection the dominant mechanism of heat transfer ? heat transfer to a piston head in a diesel engine combustion chamber heat transfer from the inside of a fan-cooled p.c. heat transfer to a solar heating panel heat transfer on the inside of a central heating panel radiator heat transfer on the outside of a central heating panel radiator 7. Which of these statements is not true ? conduction can occur in liquids conduction only occurs in solids thermal radiation can travel through empty space convection cannot occur in solids gases do not absorb thermal radiation 8. What is the heat flow through a brick wall of area 10m2, thickness 0.2m, k = 0.1 W/m K with a surface temperature on one side of 20C and 10C on the other ? 50 Watts 50 Joules 50 Watts / m2 200 Watts 200 Watts / m2 22. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 23 Introduction 9. The governing equations of fluid motion are known as: Maxwells equations C.F.D. Reynolds Stress equations Lames equations Navier Stokes equations 10. A pipe of surface area 2m2 has a surface temperature of 100C, the adjacent fluid is at 20C, the heat transfer coefficient acting between the two is 20 W/m2K. What is the heat flow by convection ? 1600 W 3200 W 20 W 40 W zero 11. The value of the Stefan-Boltzmann constant is: 56.7 x 10-6 W/m2K4 56.7 x 10-9 W/m2K4 56.7 x 10-6 W/m2K 56.7 x 10-9 W/m2K 56.7 x 10-6 W/m K Jeder Erfolg hat seine Geschichte. Made by Bosch steht fr erstklassige Qualitt eines Global Players. Bosch ist in der Schweiz an 24 Standorten vertreten. In Solothurn steht der weltweite Hauptsitz fr den Produktbereich Elektro- werkzeuge-Zubehr. Bei uns arbeiten rund 3300 Mitarbeitende aus unterschiedlichs- ten Nationen in vielfltigsten Berufen. Wir sind auch hier in der Schweiz ein internationales und global agierendes Unternehmen und bieten vielfltige Karrie- rechancen in betriebswirtschaftlichen und technischen Ttigkeitsfeldern. Entdecken Sie bei uns attraktive Karrieremglichkeiten Hier und jetzt starten Sie mit uns. www.bosch-career.ch 23. Download free ebooks at bookboon.com Heat Transfer 24 Introduction 12. Which of the following statements is true: Heat transfer by radiation . only occurs in outer space is negligible in free convection is a fluid phenomenon and travels at the speed of the fluid is an acoustic phenomenon and travels at the speed of sound is an electromagnetic phenomenon and travels at the speed of light 13. Calculate the net thermal radiation heat transfer between two surfaces. Surface A, has a temperature of 100C and Surface B, 200C. Assume they are sufficiently close so that all the radiation leaving A is intercepted by B and vice-versa. Assume also black-body behaviour. 85 W 85 W / m2 1740 W 1740 W / m2 none of these 14. The different modes of heat transfer are: forced convection, free convection and mixed convection conduction, radiation and convection laminar and turbulent evaporation, condensation and boiling cryogenic, ambient and high temperature 15. Mixed convection refers to: combined convection and radiation combined convection and conduction combined laminar and turbulent flow combined forced and free convection combined forced convection and conduction 16. The thermal diffusivity, , is defined as: = Cp / k = k Cp / = k / Cp = h L / k = L / k 24. Download free ebooks at bookboon.com Heat Transfer 25 Conduction 2. Conduction 2.1 The General Conduction Equation Conduction occurs in a stationary medium which is most likely to be a solid, but conduction can also occur in fluids. Heat is transferred by conduction due to motion of free electrons in metals or atoms in non-metals. Conduction is quantified by Fouriers law: the heat flux, q , is proportional to the temperature gradient in the direction of the outward normal. e.g. in the x-direction: dx dT qx (2.1) )/( 2 mW dx dT kqx (2.2) The constant of proportionality, k is the thermal conductivity and over an area A , the rate of heat flow in the x-direction, xQ is )(W dx dT AkQx (2.3) Conduction may be treated as either steady state, where the temperature at a point is constant with time, or as time dependent (or transient) where temperature varies with time. The general, time dependent and multi-dimensional, governing equation for conduction can be derived from an energy balance on an element of dimensions zyx ,, . Consider the element shown in Figure 2.1. The statement of energy conservation applied to this element in a time period t is that: heat flow in + internal heat generation = heat flow out + rate of increase in internal energy t T CmQQQQQQQ zzyyxxgzyx (2.4) or 0 t T CmQQQQQQQ gzzzyyyxxx (2.5) 25. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 26 Conduction As noted above, the heat flow is related to temperature gradient through Fouriers Law, so: Figure 2-1 Heat Balance for conduction in an infinitismal element Studieren in Dnemark heit: nicht auswendig lernen, sondern verstehen in Projekten und Teams arbeiten sich ausbilden in einem globalen Milieu den Professor duzen auf Englisch diskutieren Fahrrad fahren Mehr info: www.studyindenmark.dk 26. Download free ebooks at bookboon.com Heat Transfer 27 Conduction dx dT zyk dx dT AkQx dy dT zxk dy dT AkQy (2.6) dx dT yxk dz dT AkQz Using a Taylor series expansion: 3 3 3 2 2 2 !3 1 !2 1 x x Q x x Q x x Q QQ xxx xxx (2.7) For small values of x it is a good approximation to ignore terms containing x2 and higher order terms, So: x T k x zyxx x Q QQ x xxx (2.8) A similar treatment can be applied to the other terms. For time dependent conduction in three dimensions (x,y,z), with internal heat generation zyxQmWq gg /)/( 3 : t T Cq z T k zy T k yx T k x g (2.9) For constant thermal conductivity and no internal heat generation (Fouriers Equation): t T k C z T y T x T 2 2 2 2 2 2 (2.10) Where )/( Ck is known as , the thermal diffusivity (m2/s). For steady state conduction with constant thermal conductivity and no internal heat generation 02 2 2 2 2 2 z T y T x T (2.11) 27. Download free ebooks at bookboon.com Heat Transfer 28 Conduction Similar governing equations exist for other co-ordinate systems. For example, for 2D cylindrical coordinate system (r, z,). In this system there is an extra term involving 1/r which accounts for the variation in area with r. 0 1 2 2 2 2 r T rr T z T (2.12) For one-dimensional steady state conduction (in say the x-direction) 02 2 dx Td (2.13) It is possible to derive analytical solutions to the 2D (and in some cases 3D) conduction equations. However, since this is beyond the scope of this text the interested reader is referred to the classic text by Carlslaw and Jaeger (1980) A meaningful solution to one of the above conduction equations is not possible without information about what happens at the boundaries (which usually coincide with a solid-fluid or solid-solid interface). This information is known as the boundary conditions and in conduction work there are three main types: 1. where temperature is specified, for example the temperature of the surface of a turbine disc, this is known as a boundary condition of the 1st kind; 2. where the heat flux is specified, for example the heat flux from a power transistor to its heat sink, this is known as a boundary condition of the 2nd kind; 3. where the heat transfer coefficient is specified, for example the heat transfer coefficient acting on a heat exchanger fin, this is known as a boundary condition of the 3rd kind. 2.1.1 Dimensionless Groups for Conduction There are two principal dimensionless groups used in conduction. These are: The Biot number, khLBi / and The Fourier number, 2 / LtFo . It is customary to take the characteristic length scale L as the ratio of the volume to exposed surface area of the solid. The Biot number can be thought of as the ratio of the thermal resistance due to conduction (L/k) to the thermal resistance due to convection 1/h. So for Bi 1 they are not. The Fourier number can be thought of as a time constant for conduction. For 1Fo , time dependent effects are significant and for 1Fo they are not. 28. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 29 Conduction 2.1.2 One-Dimensional Steady State Conduction in Plane Walls In general, conduction is multi-dimensional. However, we can usually simplify the problem to two or even one-dimensional conduction. For one-dimensional steady state conduction (in say the x-direction): 02 2 dx Td (2.14) From integrating twice: 21 CxCT where the constants C1 and C2 are determined from the boundary conditions. For example if the temperature is specified (1st Kind) on one boundary T = T1 at x = 0 and there is convection into a surrounding fluid (3rd Kind) at the other boundary )()/( 2 fluidTThdxdTk at x = L then: fluidTT k hx TT 21 (2.15) which is an equation for a straight line. Manche haben Ihren Weg klar vor Augen. Andere lieben das Abenteuer, unbekanntes Terrain zu erkunden. Wie auch immer Sie sich Ihren Karrierepfad vorstellen bei Oliver Wyman sind Sie genau richtig. GET THERE FASTER Kontakt General Management Consulting: Natalie Bojdo, Telefon +49 89 939 49 409 Kontakt Financial Services Management Consulting: Marlene Nies, Telefon +49 69 955 120 0 www.oliverwyman.com/careers 29. Download free ebooks at bookboon.com Heat Transfer 30 Conduction To analyse 1-D conduction problems for a plane wall write down equations for the heat flux q. For example, the heat flows through a boiler wall with convection on the outside and convection on the inside: )( 1TThq insideinside ))(/( 21 TTLkq )( 2 outsideoutside TThq Rearrange, and add to eliminate T1 and T2 (wall temperatures) outsideinside otsideinside hkh TT q 111 (2.16) Note the similarity between the above equation with I = V / R (heat flux is the analogue of electrical current, temperature is of voltage and the denominator is the overall thermal resistance, comprising individual resistance terms from convection and conduction. In building services it is common to quote a U value for double glazing and building heat loss calculations. This is called the overall heat transfer coefficient and is the inverse of the overall thermal resistance. outsideinside hkh U 111 1 (2.17) 2.1.3 The Composite Plane Wall The extension of the above to a composite wall (Region 1 of width L1, thermal conductivity k1, Region 2 of width L2 and thermal conductivity k2 . . . . etc. is fairly straightforward. outsideinside otsideinside hk L k L k L h TT q 11 3 3 2 2 1 1 (2.18) 30. Download free ebooks at bookboon.com Heat Transfer 31 Conduction Example 2.1 The walls of the houses in a new estate are to be constructed using a cavity wall design. This comprises an inner layer of brick (k = 0.5 W/m K and 120 mm thick), an air gap and an outer layer of brick (k = 0.3 W/m K and 120 mm thick). At the design condition the inside room temperature is 20C, the outside air temperature is -10C; the heat transfer coefficient on the inside is 10 W/m2 K, that on the outside 40 W/m2 K, and that in the air gap 6 W/m2 K. What is the heat flux through the wall? Note the arrow showing the heat flux which is constant through the wall. This is a useful concept, because we can simply write down the equations for this heat flux. Convection from inside air to the surface of the inner layer of brick )( 1TThq inin Conduction through the inner layer of brick )(/ 21 TTLkq inin Convection from the surface of the inner layer of brick to the air gap Figure 2-2: Conduction through a plane wall 31. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 32 Conduction )( 2 gapgap TThq Convection from air gap to the surface of the outer layer of brick )( 3TThq gapgap Conduction through the outer layer of brick )(/ 43 TTLkq outout Convection from the surface of the outer layer of brick to the outside air )( 4 outout TThq The above provides six equations with six unknowns (the five temperatures T1, T2, T3, T4 and gapT and the heat flux q). They can be solved simply by rearranging with the temperatures on the left hand side. inin hqTT /)( 1 MASTER OF SCIENCE PROGRAMMES .bi.edu for more EFMD CLICK HERE for the best investment in your career My strategy class taught and enabled me to use systems and tools e.g. multvariate decision analysis in my projects. The management lectures at BI provided me with a lot of useful and applicable tools that I can adapt and use in many situations that I face in business life today. Rudiger Braun, Master of Businessand Economics, 2008 BI Norwegian School of Management (BI) offers a wide range of Master of Science (MSc) programmes in: Business & Economics Strategic Marketing Management International Marketing & Management Leadership & Organisational Psychology Political Economy Financial Economics Innovation & Entrepreneurship For more information, visit www.bi.edu/msc BI also offers programmes at Bachelor, Masters, Executive MBA and PhD levels. Visit www.bi.edu for more information. 32. Download free ebooks at bookboon.com Heat Transfer 33 Conduction inin LkqTT //)( 12 gapgap hqTT /)( 2 gapgap hqTT /)( 3 outout LkqTT //)( 43 outout hqTT /)( 4 Then by adding, the unknown temperatures are eliminated and the heat flux can be found directly outout out gapgapin in in outin hk L hhk L h TT q 1111 2 /3.27 40 1 3.0 12.0 6 1 6 1 5.0 12.0 10 1 )10(20 mWq It is instructive to write out the separate terms in the denominator as it can be seen that the greatest thermal resistance is provided by the outer layer of brick and the least thermal resistance by convection on the outside surface of the wall. Once the heat flux is known it is a simple matter to use this to find each of the surface temperatures. For example, outout ThqT /4 1040/3.274T CT 32.94 Thermal Contact Resistance In practice when two solid surfaces meet then there is not perfect thermal contact between them. This can be accounted for using an appropriate value of thermal contact resistance which can be obtained either from experimental results or published, tabulated data. 33. Download free ebooks at bookboon.com Heat Transfer 34 Conduction 2.2 One-Dimensional Steady-State Conduction in Radial Geometries: Pipes, pressure vessels and annular fins are engineering examples of radial systems. The governing equation for steady-state one-dimensional conduction in a radial system is 0 1 2 2 dr dT rdr Td (2.19) From integrating twice: 21 )ln( CxCT , and the constants are determined from the boundary conditions, e.g. if T = T1 at r = r1 and T = T2 at r = r2, then: )/ln( )/ln( 12 1 12 1 rr rr TT TT (2.20) Similarly since the heat flow )/( drdTkAQ , then for a length L (in the axial or z direction) the heat flow can be found from differentiating Equation 2.20. )/ln( 2 12 12 rr TTKL Q (2.21) To analyse 1-D radial conduction problems: Write down equations for the heat flow Q (not the flux, q, as in plane systems, since in a radial system the area is not constant, so q is not constant). For example, the heat flow through a pipe wall with convection on the outside and convection on the inside: )(2 11 TThLrQ insideinside )/ln(/)(2 1221 rrTTkLQ )(2 22 outsiedoutside TThLrQ Rearrange, and add to eliminate T1 and T2 (wall temperatures) 34. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 35 Conduction outsideinside outsideinside hrk rr hr TTL Q 2 12 1 1)/ln(1 )(2 (2.22) The extension to a composite pipe wall (Region 1 of thermal conductivity k1, Region 2 of thermal conductivity k2 . . . . etc.) is fairly straightforward. Example 2.2 The Figure below shows a cross section through an insulated heating pipe which is made from steel (k = 45 W / m K) with an inner radius of 150 mm and an outer radius of 155 mm. The pipe is coated with 100 mm thickness of insulation having a thermal conductivity of k = 0.06 W / m K. Air at Ti = 60C flows through the pipe and the convective heat transfer coefficient from the air to the inside of the pipe has a value of hi = 35 W / m2 K. The outside surface of the pipe is surrounded by air which is at 15C and the convective heat transfer coefficient on this surface has a value of ho = 10 W / m2 K. Calculate the heat loss through 50 m of this pipe. Going up in the world Youll be going up in the world after graduating from Nottingham Trent University. We are one of the best universities in England for graduate employment with over 97% of our students employed or in further study within six months of graduating.* Consistently ranked as one of the leading modern universities in The Independent Good University Guide. Courses informed by businesses and industry strong commercial partnerships with over 6,000 companies globally. Many courses include professional accreditation and / or the opportunity to undertake a work placement. This gives our graduates a competitive edge when entering the job market. An exciting student city in the middle of England. Ideally located for travel across the UK and with direct flights to destinations across Europe from East Midlands Airport. Investing 130 million across our three campuses to create an inspiring learning environment. In the most recent QAA institutional review, the University received the highest commendation for its consistent commitment to supporting students and their learning. If you would like any guidance on applying to Nottingham Trent University please contact Laura Vella, our International Officer for Europe, on Tel: +44 (0)115 848 8180 Email: laura.vella@ntu.ac.uk www.ntu.ac.uk/book 5365/07/09 * HESA DHLE 200607 35. Download free ebooks at bookboon.com Heat Transfer 36 Conduction Solution Unlike the plane wall, the heat flux is not constant (because the area varies with radius). So we write down separate equations for the heat flow, Q. Convection from inside air to inside of steel pipe )(2 11 TThLrQ inin Conduction through steel pipe )/ln(/)(2 1221 rrTTkLQ steel Conduction through the insulation )/ln(/)(2 2332 rrTTkLQ insulation Convection from outside surface of insulation to the surrounding air )(2 33 outout TThLrQ Figure 2-3: Conduction through a radial wall 36. Download free ebooks at bookboon.com Heat Transfer 37 Conduction Following the practice established for the plane wall, rewrite in terms of temperatures on the left hand side and then add to eliminate the unknown values of temperature, giving outinsulationsteelin oi hrk rr k rr hr TTL Q 3 2312 1 1)/ln()/ln(1 )(2 (2.23) 10255.0 1 06.0 )155.0/255.0ln( 45 )150.0/155.0ln( 15.035 1 )1560(502 Q WattsQ 1592 Again, the thermal resistance of the insulation is seen to be greater than either the steel or the two resistances due to convection. Critical Insulation Radius Adding more insulation to a pipe does not always guarantee a reduction in the heat loss. Adding more insulation also increases the surface area from which heat escapes. If the area increases more than the thermal resistance then the heat loss is increased rather than decreased. The so-called critical insulation radius is the largest radius at which adding more insulation will create an increase in the heat loss extinscrit kkr / Example 2.3 Find the critical insulation radius for the previous question. Solution: extinscrit hkr / 10/06.0critr mmrcrit 6 So for r3 > 6 mm, adding more insulation, as intended, will reduce the heat loss. 37. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 38 Conduction 2.3 Fins and Extended Surfaces Figure 2-4: Examples of fins (a) motorcycel engine, (b) heat sink Fins and extended surfaces are used to increase the surface area and therefore enhance the surface heat transfer. Examples are seen on: motorcycle engines, electric motor casings, gearbox casings, electronic heat sinks, transformer casings and fluid heat exchangers. Extended surfaces may also be an unintentional product of design. Look for example at a typical block of holiday apartments in a ski resort, each with a concrete balcony protruding from external the wall. This acts as a fin and draws heat from the inside of each apartment to the outside. The fin model may also be used as a first approximation to analyse heat transfer by conduction from say compressor and turbine blades. 38. Download free ebooks at bookboon.com Heat Transfer 39 Conduction 2.3.1 General Fin Equation The general equation for steady-state heat transfer from an extended surface is derived by considering the heat flows through an elemental cross-section of length x, surface area As and cross-sectional area Ac. Convection occurs at the surface into a fluid where the heat transfer coefficient is h and the temperature Tfluid. Figure 2-5 Fin Equation: heat balance on an element Writing down a heat balance in words: heat flow into the element = heat flow out of the element + heat transfer to the surroundings by convection. And in terms of the symbols in Figure 2.5 )( fluidsxxx TTAhQQ (2.24) From Fouriers Law. dx dT AkQ cx (2.25) and from a Taylors series, using Equation 2.25 x dx dT Ak dx d QQ cxxx (2.26) and so combining Equations 2.24 and 2.26 0)( fluidsc TTAhx dx dT Ak dx d (2.27) 39. Download free ebooks at bookboon.com Heat Transfer 40 Conduction The term on the left is identical to the result for a plane wall. The difference here is that the area is not constant with x. So, using the product rule to multiply out the first term on the left hand- side, gives: 0)( 1 2 2 fluid s c c c TT dx dA kA h dx dT dx dA Adx Td (2.28) The simplest geometry to consider is a plane fin where the cross-sectional area, Ac and surface area As are both uniform. Putting fluidTT and letting kAphm c/2 , where P is the perimeter of the cross-section 02 2 2 m dx d (2.29) The general solution to this is = C1emx + C2e-mx, where the constants C1 and C2, depend on the boundary conditions. It is useful to look at the following four different physical configurations: N.B. sinh, cosh and tanh are the so-called hyperbolic sine, cosine and tangent functions defined by: Convection from the fin tip ( )tipLx hh )31.2( )}(sinh)/({)(cosh )}(sinh)/({)(cosh mLkmhmL xLmkmhxLm TT TT tip tip fluidbase fluid where 0xbase hh . )32.2( )}(sinh)/({)(cosh )}(cosh)/({)(sinh )()( 2/1 mLkmhmL mLkmhmL TTAkPhQ tip tip fluidbasec Adiabatic Tip (htip = 0 in Equation 2.32) )33.2( )(cosh )(cosh mL xLm TT TT fluidbase fluid )30.2( cosh sinh tanhand 2 sinh, 2 cosh x x x ee x ee x xxxx 40. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 41 Conduction )(tanh)()( 2/1 mLTTAkPhQ fluidbasec (2.34) 3. Tip at a specified temperature (Tx = L) )(sinh )(sinh)(sinh mL xLmmx TT TT TT TT fluidbase fluidLx fluidbase fluid (2.35) )(sinh )(cosh )()( 2/1 mL TT TT mL TTAkPhQ fluidbase fluidLx fluidbasec (2.36) Infinite Fin )( xatTT fluid mx fluidbase fluid e TT TT (2.37) )()( 2/1 fluidbasec TTAkPhQ (2.38) 41. Download free ebooks at bookboon.com Heat Transfer 42 Conduction 2.3.2 Fin Performance The performance of a fin is characterised by the fin effectiveness and the fin efficiency Fin effectiveness, fin fin fin heat transfer rate / heat transfer rate that would occur in the absence of the fin )(/ fluidbasecfin TThAQ (2.39) which for an infinite fin becomes, Q given by 2/1 )/( cfin hAPk (2.40) Fin efficiency, fin fin actual heat transfer through the fin / heat transfer that would occur if the entire fin were at the base temperature. )(/ fluidbasesfin TThAQ (2.41) which for an infinite fin becomes, with Q given by Equation 2.38 2/12 )/( scfin hAPkA (2.42) Example 2.3 The design of a single pin fin which is to be used in an array of identical pin fins on an electronics heat sink is shown in Figure 2.6. The fin is made from cast aluminium, k = 180 W / m K, the diameter is 3 mm and the length 15 mm. There is a heat transfer coefficient of 30 W / m2 K between the surface of the fin and surrounding air which is at 25C. 1. Use the expression for a fin with an adiabatic tip to calculate the heat flow through a single pin fin when the base has a temperature of 55C. 2. Calculate also the efficiency and the effectiveness of this fin design. 3. How long would this fin have to be to be considered infinite ? 42. Download free ebooks at bookboon.com Heat Transfer 43 Conduction Solution For a fin with an adiabatic tip )(tanh)()( 2/1 mLTTAkPhQ fluidbasec For the above geometry mdP 3 1042.9 262 1007.74/ mdAc 224.0015.0)1007.7180/1042.930()/( 632/1 LAkhPLm c 22.0)tanh(mL 22.0)2555()1007.71801042.930( 2/163 Q WattsQ 125.0 Fin efficiency )(/ fluidbasesfin TThAQ 23 10141.0 mdLAs Figure 2-6 Pin Fin design 43. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 44 Conduction )2555(10141.030/125.0 3 fin %)5.98(985.0fin Fin effectiveness )(/ fluidbasecfin TThAQ )2555(10707.030/125.0 6 fin 6.19fin For an infinite fin, Tx = L = Tfluid. However, the fin could be considered infinite if the temperature at the tip approaches that of the fluid. If we, for argument sake, limit the temperature difference between fin tip and fluid to 5% of the temperature difference between fin base and fluid, then: 05.0 fluidb fluidLx TT TT Bernard chose a career at KPMG. He nds himself invol- ved in a diverse range of pro- jects which enable him to see behind the scenes at various listed companies. This dynamic working environment inspires him both on a professional level and through the people he works with. Inspiring careers for inspiring people. 44. Download free ebooks at bookboon.com Heat Transfer 45 Conduction Using equation 2.33 for the temperature distribution and substituting x = L, noting that cosh (0) = 1, implies that 1/ cosh (mL) = 0.05. So, mL = 3.7, which requires that L > 247 mm. Simple Time Dependent Conduction The 1-D time-dependent conduction equation is given by Equation 2.10 with no variation in the y or z directions: t T x T 1 2 2 (2.43) A full analytical solution to the 1-D conduction equation is relatively complex and requires finding the roots of a transcendental equation and summing an infinite series (the series converges rapidly so usually it is adequate to consider half a dozen terms). There are two alternative possible ways in which a transient conduction analysis may be simplified, depending on the value of the Biot number (Bi = h L /k). 2.3.3 Small Biot Number (Bi >1): Semi - Infinite Approximation When Bi is large (Bi >> 1) there are, as explained above, large temperature variations within the material. For short time periods from the beginning of the transient (or to be more precise for Fo > 1; b) Bi > 1; e) when the object is a small lump of mass 18. Under what circumstances can the semi-infinite approximation be used ? Bi >> 1; b) Bi > 1; e) when the object is at least 2m thick 19. The temperature variation with time using the lumped mass method is given by: (T Tf) / (Tinitial Tf) = e t, what is ? h A / m k h A / m L h A / m C h / m C / C 20. In free convection, the heat transfer coefficient depends on the temperature difference between surface and fluid. Which of the following statements is true ? the lumped mass equation given in Q.19 may be used without modification it is possible to use the lumped mass method, but with modification it is not possible to use the lumped mass method at all only the semi-infinite method may be used need to use CFD 57. Download free ebooks at bookboon.com Heat Transfer 58 Convection 3. Convection In Chapter 1, we introduced the three modes of heat transfer as conduction, convection and radiation. We have analysed conduction in Chapters 2 in more detail and we only used convection to provide a possible boundary condition for the conduction problem. We also described convection briefly in Chapter 1 as the energy transfer between a surface and a fluid moving over the surface. Although the mechanism of molecular diffusion contributes to convection, the dominant mechanism is the bulk motion of the fluid particles. We have also found that the conduction is dependent on the temperature gradient and thermal conductivity which is a physical property of the material. On the other hand, convection is a function of the temperature difference between the surface and the fluid and the heat transfer coefficient. The heat transfer coefficient is not a physical property of the material, but it rather depends on a number of parameters including fluid properties as well as the nature of the fluid motion. Thus to obtain an accurate measure of the convective heat transfer coefficient requires analysis of the flow pattern in the vicinity of the surface in concern. The nature of the flow motion will depend on the geometry and boundary conditions to the region of interest. Consequently, in this chapter we will develop basic methods used to characterise the flow leading to the calculation of the convective heat transfer coefficient. The concept of boundary layer will be introduced and distinction will be made between laminar and turbulent boundary layers and also the concept of transition from laminar to turbulent boundary layer will be discussed. We will also distinguish two types of flow motion leading to two distinct mechanisms of heat transfer by convection. Forced convection, where the flow is pushed against a surface by external means, such as blowing, and natural convection where flow motion is due to the action of density variations leading to flow motion caused by body forces. It is apparent from the above that convection is a complex physical phenomena governed by a large number of parameters. One way of allowing a systematic theoretical analysis is the use of the concept of dimensional analysis, which reduces the number of controlling parameters to few non-dimensional groupings. These lead to more general formulations for the convective heat transfer coefficient. 3.1 The convection equation Heat conducted through a solid must eventually find its way to the surroundings. In most cases the heat supplied to this solid comes also from its surroundings i.e. heat transferred from the air of a room to the room walls then through the walls and eventually out into the ambient air. 58. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 59 Convection This led to the introduction of Newtons equation. )( TThAQ s [W] (3.1) sT : temperature of the wall surface T : temperature of the fluid away from the wall A : heat transfer area Ts, T and A are measurable quantities. Evaluation of the convective coefficient h then completes the parameters necessary for heat transfer calculations. The convective coefficient h is evaluated in some limited cases by mathematical analytical methods and in most cases by experiments. ALEXANDER WOLLTE MEHR VERANTWORTUNG. WIR VERTRAUTEN IHM UNSERE WICHTIGSTEN KUNDEN AN. Als Alexander mit der Betreuung seiner neuen Private Banking Kunden startete, war nicht abzusehen, wohin ihn sein Engagement fhren wrde. Heute bert er 50 unserer grssten Kunden in Nahost. Seine Empfehlungen sind fr die Kunden sehr wertvoll die Erfahrung fr ihn unbezahlbar. Lesen Sie Alexanders Geschichte unter credit-suisse.com/careers LOCATION: ZURICH 59. Download free ebooks at bookboon.com Heat Transfer 60 Convection 3.2 Flow equations and boundary layer In this study, we are going to assume that the student is familiar with the flow governing equations. The interested reader can refer to basic fluid dynamics textbooks for their derivation. However, for clarity, we are going to state the flow equations to provide a more clear picture for the boundary layer simplifications mentioned below. Fluid flow is covered by laws of conservation of mass, momentum and energy. The conservation of mass is known as the continuity equation. The conservation of momentum is described by the Navier-Stokes equations while the conservation of energy is described by the energy equation. The flow equations for two dimensional steady incompressible flow in a Cartesian coordinate system are: 0 y v x u (Continuity) (3.2a) xF y u x u x p y u v x u u 2 2 2 2 (x-momentum) (3.2b) yF y v x v y p y v v x v u 2 2 2 2 (y-momentum) (3.2c) 2 2 2 2 y T x T k y T v x T ucp (Energy) (3.2d) where u and v are the flow velocities in the x and y directions, T is the temperature, p is the pressure, , and pc are the fluid density, viscosity and specific heat at constant pressure, xF and yF are the body forces in the x and y directions and is the dissipation function. 3.2.1 The velocity boundary layer Figure 3.1 shows fluid at uniform velocity U approaching a plate and the resulting development of the velocity boundary layer. When the fluid particles make contact with the surface they assume zero velocity. These particles then tend to retard the motion of particles in the fluid layer above them which in term retard the motion of particles above them and so on until at a distance y from the plate this effect becomes negligible and the velocity of the fluid particles is again almost equal to the original free stream velocityU . 60. Download free ebooks at bookboon.com Heat Transfer 61 Convection The retardation of the fluid motion which results in the boundary layer at the fluid-solid interface is a result of shear stresses ( ) acting in planes that are parallel to the fluid velocity. The shear stress is proportional to the velocity gradient and is given by dy du (3.3) The fluid flow as described above is characterized by two distinct regions: a thin fluid layer the boundary layer in which velocity gradient and shear stresses are large and an outer region the free stream where velocity gradients and shear stresses are negligible. The quantity seen in the above figure is called the boundary layer thickness. It is formally defined as the value of y at which UU 99.0 (3.4) The boundary layer velocity profile refers to the way in which the velocity u varies with distance y from the wall through the boundary layer. 3.2.2 Laminar and turbulent boundary layer In convection problems it is essential to determine whether the boundary layer is laminar or turbulent. The convective coefficient h will depend strongly on which of these conditions exists. U Free stream U Velocity boundary layer X Yx y T U Free stream U Velocity boundary layer X Yx y T x y T y x U Free stream U Velocity boundary layer X Yx y T U Free stream U Velocity boundary layer X Yx y T x y T y x Figure 0-1The velocity boundary layer on a flat plate 61. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 62 Convection There are sharp differences between laminar and turbulent flow conditions. In laminar boundary layers the fluid motion is highly ordered. Fluid particles move along streamlines. In contrast, fluid motion in the turbulent boundary layer is highly irregular. The velocity fluctuations that exist in this regular form of fluid flow result in mixing of the flow and as a consequence enhance the convective coefficient significantly. Figure 3.2 shows the flow over a flat plate where the boundary layer is initially laminar. At some distance from the leading edge fluid fluctuations begin to develop. This is the transition region. Eventually with increasing distance from the leading edge complete transition to turbulence occurs. This is followed by a significant increase in the boundary layer thickness and the convective coefficient, see Figures 3.2 and 3.4. Three different regions can be seen in the turbulent boundary layer. The laminar sublayer, the buffer layer and turbulent zone where mixing dominates. The location where transition to turbulence exists is determined by the value of theReynoldss number which is a dimensionless grouping of variables. L Re U (3.5a) Where L is the appropriate length scale. In the case of the flat plate the length is the distance x from the leading edge of the plate. Therefore, Jeder Erfolg hat seine Geschichte. Made by Bosch steht fr erstklassige Qualitt eines Global Players. Bosch ist in der Schweiz an 24 Standorten vertreten. In Solothurn steht der weltweite Hauptsitz fr den Produktbereich Elektro- werkzeuge-Zubehr. Bei uns arbeiten rund 3300 Mitarbeitende aus unterschiedlichs- ten Nationen in vielfltigsten Berufen. Wir sind auch hier in der Schweiz ein internationales und global agierendes Unternehmen und bieten vielfltige Karrie- rechancen in betriebswirtschaftlichen und technischen Ttigkeitsfeldern. Entdecken Sie bei uns attraktive Karrieremglichkeiten Hier und jetzt starten Sie mit uns. www.bosch-career.ch 62. Download free ebooks at bookboon.com Heat Transfer 63 Convection v u streamline y, v x, u Laminar Transition Turbulent x u u u Turbulent region Buffer layer Laminar sublayer v u streamline y, v x, u Laminar Transition Turbulent x u u u Turbulent region Buffer layer Laminar sublayer xU xRe (3.5b) For Rex < 5 105 the flow is laminar and for Rex > 5 105 the flow is turbulent Figure 3.2 Laminar and turbulent boundary layer on a flat plate 3.2.3 The Thermal boundary layer Figure 3.3 shows analogous development of a thermal boundary layer. A thermal boundary layer must develop similar to the velocity boundary layer if there is a difference between the fluid free stream temperature and the temperature of the plate. The fluid particles that come in contact with the plate achieve thermal equilibrium with the surface and exchange energy with the particles above them. A temperature gradient is therefore established. T T t free stream Thermal boundary layer Ts X Figure 3.3The thermal boundary layer development on a flat plate. 63. Download free ebooks at bookboon.com Heat Transfer 64 Convection The quantity t is the thickness (at any position x) of the thermal boundary layer and is formally defined as the value of y for 99.0 TT TT s s (3.6) h h , u , X T Ts Laminar Transition Turbulent At the surface of the plate and at any distance x the heat flux is given dy dT kqx where dy dT is evaluated at the wall-fluid interface and k is the conductivity of the fluid at the wall temperature. Fouriers law above can be applied because at the interface the fluid is at rest and conduction occurs. Then )( TTh dy dT kq sxx (3.7) Therefore dy dT k TT h s x 1 (3.8) Figure 3.4 Variation of the velocity boundary layer thickness and the local convective heat transfer coefficient 64. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 65 Convection Because t increases with x, the temperature gradient decrease with distance x (since the thermal boundary layer thickness increases) and therefore hx decreases with distance. Equation 3.8 indicates that to be able to calculate the heat transfer coefficient, we need to know the temperature gradient in the boundary layer. To achieve this, either accurate measurements of the temperature distribution normal to the wall are required, or the flow equations (Equations 3.2) need to be solved to obtain the temperature distribution. Analytic solutions to the flow equations are only possible for simple geometries and simple flows with various further simplifications. For most problems encountered in engineering applications, it is not possible to obtain analytic solutions to those equations. We are going to discuss the simplification to the flow equations for laminar boundary layers in the next subsection. Generally, three approaches are possible; Measurement of temperature distribution. This is a reliable approach as long as care is taken to obtain accurate measurements. However, this approach has several drawbacks. Measurements of the actual geometries might not be possible at the design stage. The measurements can be costly and time consuming. Studieren in Dnemark heit: nicht auswendig lernen, sondern verstehen in Projekten und Teams arbeiten sich ausbilden in einem globalen Milieu den Professor duzen auf Englisch diskutieren Fahrrad fahren Mehr info: www.studyindenmark.dk 65. Download free ebooks at bookboon.com Heat Transfer 66 Convection Numerical solution of the flow equation. This is becoming increasingly popular approach with the recent development of Computational Fluid Dynamics (CFD) techniques and the availability of cheap computing power. However, good knowledge of the boundary conditions is required. In addition, the accuracy is limited by the accuracy of the numerical procedure. The non-dimensionalisation procedure described in this Chapter is used with the analytical, numerical or experimental approaches mentioned above to reduce the number of analyses required. As we will see in more detail later, the controlling parameters are reduced to a much smaller number of non-dimensional groups, thus making use of the dynamic similarity in the calculation of the heat transfer coefficient. 3.2.4 Flow inside pipes As seen in Figure 3.5, when fluid with uniform velocityU enters a pipe, a boundary layer develops on the pipe surface. This development continues down the pipe until the boundary layer is thick enough so it merges at the pipe centreline. The distance it takes to merge is called hydrodynamic length. After that the flow is termed fully developed and is laminar if Red < 2300 (where Red is the Reynolds number based on the pipe diameter, d, and the fluid mean velocity inside the pipe, um, i.e. ( /Re dumd ) and turbulent if Red > 4000 with the corresponding velocity profiles as shown in the figure. If the fluid and the pipe are at different temperatures then a thermal boundary layer develops in the pipe. This is shown in Figure 3.6 below for the case Ts > Tf. The above description of the hydrodynamic and thermal behaviour of the fluid is important because the convective heat transfer coefficient depends on whether the velocity and thermal fields are developing or developed and whether the flow is laminar or turbulent. This variation is similar to the variation of the convective coefficient with varying velocity and thermal fields on a flat plate see Figure 3.3. Figure 3-5 Velocity boundary layer development in pipes (not to scale) 66. Download free ebooks at bookboon.com Heat Transfer 67 Convection 3.2.5 The boundary layer approximation Although it is possible to solve the Navier-Stokes equations analytically for simple geometries and simple flows, it is useful to make use of the nature of the boundary layer to simplify the equations before attempting to solve them. The boundary layer approximation is a simplification which recognises that the flow and temperature distribution in the boundary layers plays the most important role in affecting heat transfer. This leads to the so-called Boundary Layer Equations that are a simplified form of the Navier-Stokes equations. These arise from the simple observation, that the boundary layer thickness is much smaller than the length scale of the geometry in concern. Obviously, this is valid for a wide range of situations but not necessarily all. Using this simplification, it is possible to neglect certain terms in the flow equations based on an order of magnitude analysis (see Long 1999). For a 2-dimensional flow in the (x,y) plane, the boundary layer equations can be written as: 0 y v x u (Continuity) (Energy) (Momentum) 2 2 2 2 y T k y T v x T uc F y u dx dp y u v x u u p x These equations assume incompressible flow 0andconstant)( . This simplified form can then be solved to obtain the temperature profile near the wall and hence the heat transfer coefficient. Typically, the equations are first integrated term by term. An assumption for the shape of the velocity profile is assumed in the form of a polynomial. The Thermal entrance region Fully developed region Figure 3.6 Thermal boundary layer development on a heated tube 67. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 68 Convection coefficients of the polynomial can then be determined using the boundary conditions. This is then used to obtain the final solution. For the case of laminar flow over a flat plate, see Long (1999). It is worth mentioning here that despite the major simplifications made by the boundary layer approximation, these equations are still difficult to solve analytically and this is only possible for simple geometries and simple flows. One example is the laminar flow over a flat plate mentioned above. Thus engineers resort to numerical methods based on Computational Fluid Dynamics (CFD) for solution of more complex problems. 3.2.6 Scale analysis: Laminar forced convection for a flat plate We can further simplify the above equations using scale analysis to obtain a relation between the heat transfer coefficient and flow parameters for a laminar flow over a flat plate. This analysis serves to confirm the validity of the analytic or numerical solutions. From Equation 3.8, the heat transfer coefficient is given by: )( )/( 0 TT dydTk h s y Manche haben Ihren Weg klar vor Augen. Andere lieben das Abenteuer, unbekanntes Terrain zu erkunden. Wie auch immer Sie sich Ihren Karrierepfad vorstellen bei Oliver Wyman sind Sie genau richtig. GET THERE FASTER Kontakt General Management Consulting: Natalie Bojdo, Telefon +49 89 939 49 409 Kontakt Financial Services Management Consulting: Marlene Nies, Telefon +49 69 955 120 0 www.oliverwyman.com/careers 68. Download free ebooks at bookboon.com Heat Transfer 69 Convection Which becomes: tt k T Tk h ~ (3.9) where the symbol ~ indicates (of the same order of magnitude). If we define the heat transfer coefficient in terms of khLNu / a non-dimensional quantity called the Nusselt number, which will be elaborated upon later, then t L Nu ~ (3.10) So if we also take into account that: Lx ~ , ~y , Uu ~ Then from the continuity equation, we get: v L U ~ , or L Uv ~ (3.11) The momentum equation in terms of scale becomes: 2 2 ~, UU v L U (3.12) Substituting the estimate from Equation 3.11: 2 22 ~, U L U L U (3.13) From Equation 3.13, we notice that each of the inertial terms (on the left hand side) of the momentum equation are of comparable magnitude, hence: 2 2 ~ U L U (3.14) Rearranging: we get: 69. Download free ebooks at bookboon.com Heat Transfer 70 Convection 2/1 2 1 Re~ L LUL (3.15) Exact analysis of the equations shows that: 2/1 Re64.4 L x (3.16) From a similar order of magnitude analysis for the energy equation, we obtain the following relation for oils and gases (but not for fluids with Pr 1 and the opposite is true. From the above interpretation it follows that the Pr number affects the growth of the velocity and the thermal boundary layers, i.e. in laminar flow number)positiveais(Pr knesslayer thicboundaryThermal knesslayer thicboundaryVelocity nn Equation 3.25 forms the basis for general formulations for the non-dimensional heat transfer coefficient as a function of the Reynolds number and the Prandtl numbers as follows: ba Nu (Pr)(Re) or ba CNu (Pr)(Re) (3.26) The constants C, a and b can be obtained either by experiments, numerical methods or from analytic solutions if they were possible. In the following subsection we will present the values of these coefficients for common geometries encountered in engineering applications. It is worth at this juncture to compare Equation 3.26 with Equation 3.19, where the constants C, a and b are given as 0.644, 1/2 and 1/3 for laminar flow over a flat plate. 3.4 Forced Convection relations In this section, we will present forced convection relations for various geometric situations encountered in engineering applications. These are either analytic or empirical relations (obtained from experiments. These will be supported by worked examples to help the student in understanding the application of these relations. 3.4.1 Laminar flow over a flat plate In the simple case of an isothermal flat plate, as mentioned earlier, analytical methods can be used to solve either the Navier-Stokes equations or the boundary layer approximation (section 3.2.5) for Nux. This then can be integrated to obtain an overall Nusselt number NuL for the total heat transfer from the plate. The derivation of the analytic solution for the boundary layer equations can be found in Long (1999). This leads to the following solutions: 2 1 3 1 332.0 xu k c k xh px (3.27) or 2 1 3 1 RePr332.0 xxNu (3.28) 76. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 77 Convection The above physical properties (k, pc , , ) of the fluid can vary with temperature and are therefore evaluated at the mean film temperature 2/TTT sfilm )( Where sT is the plate is surface temperature and T is the temperature of the fluid in the farstream. Equation 3.28 is applicable for Rex < 5x105 i.e. laminar flow and Pr 0.6 (air and water included). In most engineering calculations, an overall value of the heat transfer coefficient is required rather than the local value. This can be obtained by integrating the heat transfer coefficient over the plate length as follows: )()(element)of(area TTdxbTThq ssxx (3.29) L dx x Plate width = b 77. Download free ebooks at bookboon.com Heat Transfer 78 Convection And for the plate entire length )( TThLbqQ sL L x 0 (3.30) Where Lh is the average convective coefficient andQ is the total heat transfer from the plate. If we substitute for xq from Equation 3.29 and rearrange we have, L xL dxh L h 0 1 (3.31) If we now substitute for xh from Equation 3.27 and integrate the resulting equation the average heat transfer coefficient is 2/13/1 RePr664.0 L L k Lh (3.32) or 2/13/1 RePr664.0 LLNu (3.33) This is valid for ReL < 5x105 Comparing Equation 3.28 with Equation 3.33 it can be seen that Lxat2 hhL Figure 3.7 shows a situation where there is an initial length of the flat plate which is not heated. The velocity boundary layer starts at 0x while the thermal boundary layer starts at oxx . In this case, it is possible to modify Equation 3.28 as follows: 78. Download free ebooks at bookboon.com Heat Transfer 79 Convection 3/14/3 0 3/12/1 1 PrRe332.0 x x Nu x (3.34) 3.4.2 Turbulent flow over a flat plate Transition from laminar to turbulent flow over a flat plate generally takes place at a Reynolds number of approximately 5 105 . This value however can vary up to an order of magnitude either way dependent on the state of free stream turbulence and the smoothness of the plate. Figure 3.8 shows a schematic of the velocity profile near the wall for turbulent and laminar flow It is apparent that the velocity gradient is much steeper for turbulent flow. By inspecting Equation 3.8, it is clear that for the same temperature difference between the wall and the free stream flow, turbulent flow will produce a larger heat transfer coefficient. Physically, this can be explained by the fact that there is more mixing of the flow due to turbulent fluctuations leading to a higher heat transfer. Figure 3-7: Thermal boundary layer starting at a different position from momentum boundary layer Figure 0-8 Velocity profiles for laminar and turbulent flows 79. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 80 Convection For a flat plate, the Universal shear stress relationship for the velocity profile can be used, which is also known as the one-seventh power law 7/1 )(7.8 yu (3.35) Which gives a good fit for experimental measurements. Where * U u u , yU y * and s U * is know as the shear stress velocity. Reynolds made an analogy between the momentum equation and energy equation, which lead to an expression of the Nusselt number as a function of the Reynolds number and wall shear stress as follows: fxxx CNu Re2 1 (3.36) where 2 2 1 U C s fx (3.37) 80. Download free ebooks at bookboon.com Heat Transfer 81 Convection Equation 3.36 was obtained by imposing a condition of Pr = 1. However, since all fluids have a Pr which is different from unity, this equation was modified using empirical data using a factor of 3/1 Pr which gives: 3/1 2 1 PrRe fxxx CNu (3.38) From Equation 3.35: 7/1* * 7.8 U U U (3.39) Dividing 3.35 by 3.39: 7/1 y U u (3.40) Also, since 2/1* /sU then rearranging Equation 3.39 gives: 4/1 4/7 0255.0 Us (3.41) To obtain an expression for the shear stress, we need information about the growth of the boundary layer thickness . This can be found by integrating the momentum boundary layer Equation (see Long 1999) to obtain: 2.0 Re37.0 x x (3.42) Substituting into Equation 3.36 gives: 2.0 Re0576.0 xfxC (3.43) Substituting into Equation 3.38 gives: 3/18.0 PrRe0288.0 xxNu (3.44) 81. Download free ebooks at bookboon.com Heat Transfer 82 Convection To compute the average Nusselt number for turbulent flow, it should be taken into account the a proportion of the blade will have a laminar boundary layer followed by transition before the flow becomes turbulent. So ideally, the Nusselt number needs to be integrated along the wall taking into account the change in the nature of the boundary layer. This leads to the following formula for the average Nusselt number (See Incropera and DeWitt 2002): 3/18.0 , 8.05.0 , PrReRe037.0Re664.0 cxLcxLNu (3.45) Where cx,Re is the Reynolds number at which transition occurs. Assuming that transition occurs at 5 105ReL , Equation 3.45 becomes: 3/18.0 Pr871Re037.0 LLNu (3.46) If the length at which transition occurs is much smaller than the total length L of the plate, then Equation 3.46 can be approximated by 3/18.0 PrRe037.0 LLNu (3.47) Example 3.1 Air at temperature 527oC and 1 bar pressure flows with a velocity of 10m/s over a flat plate 0.5m long. Estimate the cooling rate per unit width of the plate needed to maintain it at a surface temperature of 27oC assuming the contribution of radiation contribution is negligible. Solution: The first step in to compute the heat transfer coefficient. This requires information allowing the computation of the Reynolds and Prandtl numbers. For a large temperature difference between surface and fluid, the properties of air are evaluated at the mean film temperature (Section 3.4.1): From tabulated data, we obtain the following properties of air at 277oC and 1 bar: 25 /10884.2 mNs 3 /6329.0 mkg kgKJcp /1040 mKWk /044.0 CTfilm 277)52727(5.0 82. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 83 Convection The Reynolds number based on the plate length is then computed as: 109726 10884.2 5.0106329.0 Re 5 LU L 5 105ReL This means that the flow is laminar along the plate length, so we can use Equation 3.33 to calculate the average Nusselt number. 681.0 044.0 104010884.2 Pr 5 k cp 5.193109726681.0664.0RePr664.0 2/13/12/13/1 LNu KmW L kNu h L 2 /17 5.0 044.05.193 WTThAQ s 425027527)15.0(17)( . This is the amount of heat that needs to be removed from the surface to keep it at a constant temperature of 27oC. Bernard chose a career at KPMG. He nds himself invol- ved in a diverse range of pro- jects which enable him to see behind the scenes at various listed companies. This dynamic working environment inspires him both on a professional level and through the people he works with. Inspiring careers for inspiring people. 83. Download free ebooks at bookboon.com Heat Transfer 84 Convection Example 3.2 A square flat plate of 2m each side is maintained at a uniform temperature of 230 oC by means of an embedded electric wire heater. If the atmospheric air is flowing at 25 oC over the plate with a velocity of 60m/s, what is the electrical power input required? Solution: Properties of air need to be evaluated at the mean film temperature of: CTfilm 5.127)25275(5.0 From tabulated data, we obtain the following properties of air at 150oC (400K): 25 /10301.2 mNs 3 /871.0 mkg kgKJcp /1014 mKWk /0338.0 5 5 10422.45 10301.2 260871.0 Re LU L 5 105ReL This means that the flow is turbulent over most of the plate length. Assuming that transition occurs at 5 105ReL , this means it will start at 0.22 m from the leading edge which is a small, but not negligible portion of the plate (just over 10%). We will use both Equations 3.46 and 3.47 to estimate the error resulting from the approximation in Equation 3.47 690.0 0338.0 101310301.2 Pr 5 k cp From Equation 3.46 6153690.087110422.45037.0Pr871Re037.0 3/18.053/18.0 LLNu 6923690.010422.45036.0PrRe036.0 3/18.053/18.0 LLNu Percentage error %5.12100 6153 61536923 Note that this error will reduce the increasing ReL. 84. Download free ebooks at bookboon.com Heat Transfer 85 Convection Thus KmW L kNu h L 2 /104 0.2 0338.06153 WTThAQ s 8528025230)22(104)( Thus 75.28 kW of electric power needs to be supplied to keep the plate temperature. 3.4.3 Laminar flow in pipes In Section 3.2.4, we discussed the nature of the flow in pipes and distinguished between two types of fully developed pipe flow, laminar and turbulent. If the Reynolds number dRe , based on the pipe diameter is less than about 2300, then the flow is laminar. For Reynolds number above 2300 the boundary layer developing at the entrance of the pipe undergoes transition and becomes turbulent leading to a fully developed turbulent flow. For laminar pipe flow, the velocity and temperature profiles can be derived from the solution of the flow Naveir-Stokes equations or boundary layer approximations leading to the determination of the heat transfer coefficient. We will not go it detailed derivations here and the interested reader can refer to Long (1999) In pipe flow, we seek to determine a heat transfer coefficient such that Newtons law of cooling is formulated as: )( ms TThq (3.48) Where sT is the pipe-wall temperature and mT is the mean temperature in the fully developed profile in the pipe. The mean fluid temperature is used instead of the free stream temperature for external flow. In a similar fashion to the average Nusselt number for a flat plate, we define the average Nusselt number for pipe flow as: kdhNud / , were d is the pipe diameter. From the derivation of formulae for the average Nusselt number for laminar follow resulting from the solution of the flow equations, it turns out the the Nusselt number is constant, and does not depend on either the Reyonlds or Prandtl numbers so long as the Reynolds number is below 2300. However, two different solutions are found depending on the physical situation. For constant heat flux pipe flow, the Nusselt number is given by (see Incropera and DeWitt 2002): 85. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 86 Convection 36.4dNu (3.49) While for a pipe with constant wall temperature, it is given by: 66.3dNu (3.50) Note that to determine h from Equations 3.49 and 3.50, the heat transfer coefficient needs to be determined at the mean temperature mT . For pipes were there is a significant variation of temperature between entry and exit of the pipe (such as in heat exchangers), then the fluid properties need to be determined at the arithmetic mean temperature between entry and exit (i..e at 2/)( exitmentrym TT ). 3.4.4 Turbulent flow in pipes Determination of the heat transfer coefficient for turbulent pipe flow analytically is much more involved than that for laminar flow. Hence, greater emphasis is usually placed on empirical correlations. The classic expression for local Nusselt number in turbulent pipe flow is due to Colburn, which is given by: Bereit fr neue Wege? Nordafrika Bestimmung einer neuen Landwirtschaftsstrategie fr ein Entwicklungsland. Herausforderungen gab es viele, sowohl in wirtschaftlicher, sozialer, ko- logischer als auch in politischer Hinsicht. McKin- sey hat Lsungen erarbeitet, die das Wachstum der Landwirtschaft dank Produkten sichern, fr die eine grosse Nachfrage besteht und die hohe Gewinne einbringen. Ausserdem wurden lokale Initiativen fr den Kampf gegen die Armut in lndlichen Gegenden ins Leben gerufen. Schliessen Sie sich uns an. www.mckinsey.ch creatives.com 86. Download free ebooks at bookboon.com Heat Transfer 87 Convection 3/25/4 PrRe0.023 ddNu (3.51) However, it is found that the Dittus-Boelter equation below provides a better correlation with measured data: n ddNu PrRe0.023 5/4 (3.52) where 4.0n for heating (Ts > Tm) and 3.0n for cooling (Ts < Tm) Equation 3.52 is valid for: Red > 104, L / D > 10 CTT fs 5 for liquids and CTT fs 55 for gasses For larger temperature differences use of the following formula is recommended (Sieder and Tate, 1936). 14.0 315/4 PrRe027.0 s / ddNu (3.53) For 16700Pr7.0 , 000,10Red and 10 d L Where s is the viscosity evaluated at the pipe surface temperature. The rest of the parameters are evaluated at the mean temperature. Example 3.3 A concentric pipe heat exchanger is used to cool lubricating oil for a large diesel engine. The inner pipe of radius 30mm and has water flowing at a rate of 0.3 kg/s. The oil is flowing in the outer pipe, which has a radius of 50mm, at a rate of 0.15kg/s. Assuming fully developed flow in both inner and outer pipes, calculate the heat transfer coefficient for the water and oil sides respectively. Evaluate oil properties at 80oC and water properties at 35oC. 87. Download free ebooks at bookboon.com Heat Transfer 88 Convection Solution The first step is to obtain the relevant data from tables: For oil at 80oC: 2131pc J/kg K, 2 1025.3 Kg/m s, 138.0k W/m K For water at 35oC: 4178pc J/kg K, 6 10725 Kg/m s, 625.0k W/m K The next step is to evaluate the Reynolds numbers: VAm 4 2 d m A m V d m d dmVd d 44 Re 2 For water: 8781 1072506.0 3.04 Re 6 85.4 625.0 418710725 Pr 6 k cp 88. Download free ebooks at bookboon.com Pleaseclicktheadvert Heat Transfer 89 Convection Since Re > 2300, the flow is turbulent. Thus we can use Equation 3.52, with the exponent 0.4 as the water is being heated by the oil. Thus 6285.48781023.0PrRe023.0 4.08.04.08.0 ddNu and 643 06.0 625.062 d kNu h d W/m2 K Oil is flowing in an annular shape pipe. We can use the same relations as a circular pipe, however, we use the hydraulic diameter instead of the diameter for calculating the Reynolds number. The hydraulic diameter is defined as: perimeterWetter Area4 hd 89. Download free ebooks at bookboon.com Heat Transfer 90 Convection Thus 036.0)03.005.0(2)(2 )(2 )(4 22 io io io h rr rr rr d m )( 22 io rr m A m V )( 2 )( )(2 Re 22 ioio ioh d rr m rr rrmVd 36 )03.005.0(1025.3 15.02 Re 2d Since Re < 2300, the flow laminar. Thus we can use Equation 3.49, which gives a Nusselt number of 3.36 for pipe flow assuming a wall with a constant heat flux. Then 7.16